135 citations · 516 across the 43 of their papers we have counts for
33 papers · 1 filter
Scary Wheels (and Super Shrubs)
Andrei Neguţ, Alexander Tsymbaliuk
We prove the shuffle realization for quantum affine superalgebras of types in the loop realization, in that we construct an isomorphism between each of these algebras and a…
Chains of affine standard Lyndon words
Corbet Elkins, Alexander Tsymbaliuk
In this note, we establish the periodicity of chains of affine standard Lyndon words in all types and determine tight bounds on that periodicity, greatly generalizing the -type…
Orthosymplectic quantum groups revisited
Kyungtak Hong, Alexander Tsymbaliuk
We present the RLL-realization of extended orthosymplectic quantum supergroups for any parity sequence, with R-matrices evaluated in the earlier work arxiv:2408.16720. Our isomorph…
On De Concini-Kac forms of quantum groups
Ivan Losev, Alexander Tsymbaliuk, Trung Vu
Quantum groups of semisimple Lie algebras at roots of unity admit several different forms. Among them is the De Concini-Kac form, which is the easiest to define but, perhaps, harde…
Bicharacter twists of quantum groups
Ian Martin, Alexander Tsymbaliuk
We apply the general construction of a twist of bigraded Hopf algebras by skew bicharacters to obtain two-parameter quantum groups in the Drinfeld-Jimbo, new Drinfeld (for affine t…
Affine standard Lyndon words
Corbet Elkins, Alexander Tsymbaliuk
In this note, we establish the convexity and monotonicity for affine standard Lyndon words in all types, generalizing the -type results of arXiv:2305.16299. We also derive parti…
Orthogonal bases for two-parameter quantum groups
Ian Martin, Alexander Tsymbaliuk
In this note, we construct dual PBW bases of the positive and negative subalgebras of the two-parameter quantum groups in classical types, as used in our ea…
Orthosymplectic -matrices
Kyungtak Hong, Alexander Tsymbaliuk
We present a formula for trigonometric orthosymplectic -matrices associated with any parity sequence, and establish their factorization into the ordered product of -exponents…
Fusion and specialization for type ADE shuffle algebras
Andrei Neguţ, Alexander Tsymbaliuk
Root vectors in quantum groups (of finite type) generalize to fused currents in quantum loop groups ([5]). In the present paper, we construct fused currents as duals to specializat…
Standard Lyndon loop words: weighted orders
Severyn Khomych, Nazar Korniichuk, Kostiantyn Molokanov +1
We generalize the study of standard Lyndon loop words from [A.Negut, A.Tsymbaliuk, "Quantum loop groups and shuffle algebras via Lyndon words", Adv. Math. 439 (2024), Paper No. 109…
Two-Parameter Quantum Groups and -Matrices: Classical Types
Ian Martin, Alexander Tsymbaliuk
We construct finite -matrices for the first fundamental representation of two-parameter quantum groups for classical , both through the…
Orthosymplectic Yangians
Rouven Frassek, Alexander Tsymbaliuk
We study the RTT orthosymplectic super Yangians and present their Drinfeld realizations for any parity sequence, generalizing the results for non-super types BCD, a standard parity…
Affine Standard Lyndon words: A-type
Yehor Avdieiev, Alexander Tsymbaliuk
We generalize an algorithm of Leclerc describing explicitly the bijection of Lalonde-Ram from finite to affine Lie algebras. In type , we compute all affine standard Lyn…
Shuffle approach towards quantum affine and toroidal algebras
Alexander Tsymbaliuk
These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of Marc…
Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems
Alexander Tsymbaliuk
We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras, thus generalizing its rational version of arXiv:2104.1451…
Transfer matrices of rational spin chains via novel BGG-type resolutions
Rouven Frassek, Ivan Karpov, Alexander Tsymbaliuk
We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras , whose highest weight is a multiple…
Rational Lax matrices from antidominantly shifted extended Yangians: BCD types
Rouven Frassek, Alexander Tsymbaliuk
Generalizing our recent joint paper with Vasily Pestun (arXiv:2001.04929), we construct a family of rational Lax matrices, polynomial in the spectral param…
Quantum loop groups and shuffle algebras via Lyndon words
Andrei Neguţ, Alexander Tsymbaliuk
We study PBW bases of the untwisted quantum loop group (in the Drinfeld new presentation) using the combinatorics of loop words, by generalizing the treatment…
Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type
Rouven Frassek, Vasily Pestun, Alexander Tsymbaliuk
We construct a family of rational and trigonometric Lax matrices parametrized by -valued divisors on . To this end, we study the shifted Drin…
Shuffle algebra realizations of type super Yangians and quantum affine superalgebras for all Cartan data
Alexander Tsymbaliuk
We introduce super Yangians of (in the new Drinfeld realization) associated to all Dynkin diagrams of , where is a finite-…
Duality of Lusztig and RTT integral forms of
Alexander Tsymbaliuk
We show that the Lusztig integral form is dual to the RTT integral form of the type A quantum loop algebra with respect to the new Drinfeld pairing, by utilizing the shuffle algebr…
Shifted quantum affine algebras: integral forms in type (with appendices by Alexander Tsymbaliuk and Alex Weekes)
Michael Finkelberg, Alexander Tsymbaliuk
We define an integral form of shifted quantum affine algebras of type and construct Poincaré-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form…
PBWD bases and shuffle algebra realizations for and their integral forms
Alexander Tsymbaliuk
We construct a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the quantum loop algebras …
On Sevostyanov's construction of quantum difference Toda lattices
Roman Gonin, Alexander Tsymbaliuk
We propose a natural generalization of the construction of the quantum difference Toda lattice (introduced independently by Etingof and Sevostyanov) associated to a simple Lie alge…